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Multi-body wave function of ground and low-lying excited states using unornamented deep neural networks

Tomoya Naito (内藤智也)1,*, Hisashi Naito (内藤久資)2,†, and Koji Hashimoto (橋本幸士)3,‡

  • 1RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), Wako 351-0198, Japan and Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
  • 2Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
  • 3Department of Physics, Kyoto University, Kyoto 606-8502, Japan
  • *tnaito@ribf.riken.jp
  • †naito@math.nagoya-u.ac.jp
  • ‡koji@scphys.kyoto-u.ac.jp
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Phys. Rev. Research 5, 033189 – Published 14 September, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.033189

Abstract

We propose a method to calculate wave functions and energies not only of the ground state but also of low-lying excited states using a deep neural network and the unsupervised machine learning technique. For systems composed of identical particles, a simple method to perform symmetrization for bosonic systems and antisymmetrization for fermionic systems is also proposed.

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References (83)

  1. L. D. Faddeev, Scattering theory for a three-particle system, Sov. Phys. JETP 12, 1014 (1961) [Zh. Eksp. Teor. Fiz. 39, 1459 (1961)].
  2. O. A. Yakubovskĭ, On the integral equations in the theory of N particle scattering, Sov. J. Nucl. Phys. 5, 937 (1967) [J. Nucl. Phys. (U.S.S.R.) 5, 1312 (1967)].
  3. M. Fabre de la Ripelle, The potential harmonic expansion method, Ann. Phys. 147, 281 (1983).
  4. J. Carlson, Green's function Monte Carlo study of light nuclei, Phys. Rev. C 36, 2026 (1987).
  5. M. Kamimura, Nonadiabatic coupled-rearrangement-channel approach to muonic molecules, Phys. Rev. A 38, 621 (1988).
  6. H. Kamada and W. Glöckle, Solutions of the Yakubovsky equations for four-body model systems, Nucl. Phys. A 548, 205 (1992).
  7. K. Varga and Y. Suzuki, Precise solution of few-body problems with the stochastic variational method on a correlated Gaussian basis, Phys. Rev. C 52, 2885 (1995).
  8. M. Viviani, A. Kievsky, and S. Rosati, Calculation of the α-particle ground state, Few-Body Syst. 18, 25 (1995).
  9. P. Navrátil and B. R. Barrett, Four-nucleon shell-model calculations in a Faddeev-like approach, Phys. Rev. C 59, 1906 (1999).
  10. P. Navrátil, G. P. Kamuntavičius, and B. R. Barrett, Few-nucleon systems in a translationally invariant harmonic oscillator basis, Phys. Rev. C 61, 044001 (2000).
  11. N. Barnea, W. Leidemann, and G. Orlandini, State dependent effective interaction for the hyperspherical formalism, Phys. Rev. C 61, 054001 (2000).
  12. H. Kamada, A. Nogga, W. Glöckle, E. Hiyama, M. Kamimura, K. Varga, Y. Suzuki, M. Viviani, A. Kievsky, S. Rosati, J. Carlson, S. C. Pieper, R. B. Wiringa, P. Navrátil, B. R. Barrett, N. Barnea, W. Leidemann, and G. Orlandini, Benchmark test calculation of a four-nucleon bound state, Phys. Rev. C 64, 044001 (2001).
  13. D. Ceperley, Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions, Phys. Rev. B 18, 3126 (1978).
  14. D. M. Ceperley and B. J. Alder, Ground State of the Electron Gas by a Stochastic Method, Phys. Rev. Lett. 45, 566 (1980).
  15. J. J. Shepherd, G. Booth, A. Grüneis, and A. Alavi, Full configuration interaction perspective on the homogeneous electron gas, Phys. Rev. B 85, 081103(R) (2012).
  16. J. J. Shepherd, G. H. Booth, and A. Alavi, Investigation of the full configuration interaction quantum Monte Carlo method using homogeneous electron gas models, J. Chem. Phys. 136, 244101 (2012).
  17. G. H. Booth, A. Grüneis, G. Kresse, and A. Alavi, Towards an exact description of electronic wave functions in real solids, Nature (London) 493, 365 (2013).
  18. D. Lonardoni, S. Gandolfi, J. E. Lynn, C. Petrie, J. Carlson, K. E. Schmidt, and A. Schwenk, Auxiliary field diffusion Monte Carlo calculations of light and medium-mass nuclei with local chiral interactions, Phys. Rev. C 97, 044318 (2018).
  19. T. Shen, Y. Liu, Y. Yu, and B. M. Rubenstein, Finite temperature auxiliary field quantum Monte Carlo in the canonical ensemble, J. Chem. Phys. 153, 204108 (2020).
  20. J. A. Pople, J. S. Binkley, and R. Seeger, Theoretical models incorporating electron correlation, Int. J. Quantum Chem. 10, 1 (1976).
  21. J. A. Pople, R. Seeger, and K. Raghavachari, Variational configuration interaction methods and comparison with perturbation theory, Int. J. Quantum Chem. 12, 149 (1977).
  22. J. A. Pople, Nobel Lecture: Quantum chemical models, Rev. Mod. Phys. 71, 1267 (1999).
  23. F. Coester, Bound states of a many-particle system, Nucl. Phys. 7, 421 (1958).
  24. F. Coester and H. Kümmel, Short-range correlations in nuclear wave functions, Nucl. Phys. 17, 477 (1960).
  25. J. Čížek, On the correlation problem in atomic and molecular systems: Calculation of wave-function components in Ursell-type expansion using quantum-field theoretical methods, J. Chem. Phys. 45, 4256 (1966).
  26. J. Čížek and J. Paldus, Correlation problems in atomic and molecular systems III. Rederivation of the coupled-pair many-electron theory using the traditional quantum chemical methodst, Int. J. Quantum Chem. 5, 359 (1971).
  27. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  28. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  29. W. Kohn, Nobel Lecture: Electronic structure of matter—Wave functions and density functionals, Rev. Mod. Phys. 71, 1253 (1999).
  30. S. R. White, Density Matrix Formulation for Quantum Renormalization Groups, Phys. Rev. Lett. 69, 2863 (1992).
  31. S. R. White and R. L. Martin, Ab initio quantum chemistry using the density matrix renormalization group, J. Chem. Phys. 110, 4127 (1999).
  32. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  33. A. Baiardi and M. Reiher, The density matrix renormalization group in chemistry and molecular physics: Recent developments and new challenges, J. Chem. Phys. 152, 040903 (2020).
  34. V. I. Anisimov, A. I. Poteryaev, M. A. Korotin, A. O. Anokhin, and G. Kotliar, First-principles calculations of the electronic structure and spectra of strongly correlated systems: Dynamical mean-field theory, J. Phys.: Condens. Matter 9, 7359 (1997).
  35. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic structure calculations with dynamical mean-field theory, Rev. Mod. Phys. 78, 865 (2006).
  36. G. Knizia and G. K.-L. Chan, Density Matrix Embedding: A Simple Alternative to Dynamical Mean-Field Theory, Phys. Rev. Lett. 109, 186404 (2012).
  37. R. Brockmann and J. Frank, Lattice Quantum Hadrodynamics, Phys. Rev. Lett. 68, 1830 (1992).
  38. J.-W. Chen and D. B. Kaplan, Lattice Theory for Low-Energy Fermions at Nonzero Chemical Potential, Phys. Rev. Lett. 92, 257002 (2004).
  39. D. Lee, Lattice simulations for few- and many-body systems, Prog. Part. Nucl. Phys. 63, 117 (2009).
  40. J. E. Drut and A. N. Nicholson, Lattice methods for strongly interacting many-body systems, J. Phys. G 40, 043101 (2013).
  41. L. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, NY, 1968).
  42. A. D. Becke, Perspective: Fifty years of density-functional theory in chemical physics, J. Chem. Phys. 140, 18A301 (2014).
  43. R. Jastrow, Many-body problem with strong forces, Phys. Rev. 98, 1479 (1955).
  44. P. Seth, P. L. Ríos, and R. J. Needs, Quantum Monte Carlo study of the first-row atoms and ions, J. Chem. Phys. 134, 084105 (2011).
  45. Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada, Restricted Boltzmann machine learning for solving strongly correlated quantum systems, Phys. Rev. B 96, 205152 (2017).
  46. D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes, Ab initio solution of the many-electron Schrödinger equation with deep neural networks, Phys. Rev. Res. 2, 033429 (2020).
  47. J. Hermann, Z. Schätzle, and F. Noé, Deep-neural-network solution of the electronic Schrödinger equation, Nat. Chem. 12, 891 (2020).
  48. G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2, 303 (1989).
  49. K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4, 251 (1991).
  50. T. Cheon, Green function Monte Carlo method for excited states of quantum system, Prog. Theor. Phys. 96, 971 (1996).
  51. H. Masui and T. Sato, Monte Carlo study of bound states in a few-nucleon system: Method of continued fractions, Prog. Theor. Phys. 100, 977 (1998).
  52. L. Hedin, On correlation effects in electron spectroscopies and the GW approximation, J. Phys.: Condens. Matter 11, R489 (1999).
  53. M. Bender, P.-H. Heenen, and P.-G. Reinhard, Self-consistent mean-field models for nuclear structure, Rev. Mod. Phys. 75, 121 (2003).
  54. G. Colò, P. F. Bortignon, H. Sagawa, K. Moghrabi, M. Grasso, and N. Van Giai, Microscopic theory of particle-vibration coupling, J. Phys.: Conf. Ser. 321, 012018 (2011).
  55. T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, and K. Yabana, Time-dependent density-functional description of nuclear dynamics, Rev. Mod. Phys. 88, 045004 (2016).
  56. L. Reining, The GW approximation: Content, successes and limitations, WIREs Comput. Mol. Sci. 8, e1344 (2018).
  57. M. T. Entwistle, Z. Schätzle, P. A. Erdman, J. Hermann, and F. Noé, Electronic excited states in deep variational Monte Carlo, Nat. Commun. 14, 274 (2023).
  58. H. Saito, Method to solve quantum few-body problems with artificial neural networks, J. Phys. Soc. Jpn. 87, 074002 (2018).
  59. J. W. T. Keeble and A. Rios, Machine learning the deuteron, Phys. Lett. B 809, 135743 (2020).
  60. Y. Wang, Y. Liao, and H. Xie, Solving Schrödinger equation using tensor neural network, arXiv:2209.12572 [physics.comp-ph].
  61. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg et al., TensorFlow: Large-scale machine learning on heterogeneous systems (2015), software available from tensorflow.org.
  62. D. P. Kingma and J. Ba, A method for stochastic optimization, in Third International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA,May 7–9, 2015, Conference Track Proceedings, edited by Y. Bengio and Y. Le-Cun (2015), arXiv:1412.6980 [cs.LG].
  63. N. Tajima, Hartree-Fock+BCS Approach to Unstable Nuclei with the Skyrme Force, Prog. Theor. Phys. Suppl. 142, 265 (2001).
  64. If H̃ is directly discretized, wave functions of the ground state and (M−2) excited states can be obtained.
  65. Due to the internal code of the Tensorflow, when the error of the loss function is calculated, a single precision floating point number (float32) seems to be used. Thus, the error of the loss function becomes zero if its actual value is smaller than about 1.0×10−7. Note that the convergence criteria of the DFT calculation is often about 1.0×10−8 or even 1.0×10−10; for instance, see the sample input of Ref. [83].
  66. K. Kisamori, S. Shimoura, H. Miya, S. Michimasa, S. Ota, M. Assie, H. Baba, T. Baba, D. Beaumel, M. Dozono, T. Fujii, N. Fukuda, S. Go, F. Hammache, E. Ideguchi, N. Inabe, M. Itoh, D. Kameda, S. Kawase, T. Kawabata et al., Candidate Resonant Tetraneutron State Populated by the 4He(8He,8Be) Reaction, Phys. Rev. Lett. 116, 052501 (2016).
  67. T. Faestermann, A. Bergmaier, R. Gernhäuser, D. Koll, and M. Mahgoub, Indications for a bound tetraneutron, Phys. Lett. B 824, 136799 (2022).
  68. M. Duer, T. Aumann, R. Gernhäuser, V. Panin, S. Paschalis, D. M. Rossi, N. L. Achouri, D. Ahn, H. Baba, C. A. Bertulani, M. Böhmer, K. Boretzky, C. Caesar, N. Chiga, A. Corsi, D. Cortina-Gil, C. A. Douma, F. Dufter, Z. Elekes, J. Feng et al., Observation of a correlated free four-neutron system, Nature (London) 606, 678 (2022).
  69. A. N. Wenz, G. Zürn, S. Murmann, I. Brouzos, T. Lompe, and S. Jochim, From few to many: Observing the formation of a Fermi sea one atom at a time, Science 342, 457 (2013).
  70. K. Choo, G. Carleo, N. Regnault, and T. Neupert, Symmetries and Many-Body Excitations with Neural-Network Quantum States, Phys. Rev. Lett. 121, 167204 (2018).
  71. Y. Nomura, Machine learning quantum states—Extensions to Fermion–Boson coupled systems and excited-state calculations, J. Phys. Soc. Jpn. 89, 054706 (2020).
  72. S. A. Sato, K. Yabana, Y. Shinohara, T. Otobe, K.-M. Lee, and G. F. Bertsch, Time-dependent density functional theory of high-intensity short-pulse laser irradiation on insulators, Phys. Rev. B 92, 205413 (2015).
  73. Y. Tanaka and S. Tsuneyuki, Development of the temperature-dependent interatomic potential for molecular dynamics simulation of metal irradiated with an ultrashort pulse laser, J. Phys.: Condens. Matter 34, 165901 (2022).
  74. K. Yabana, T. Tazawa, Y. Abe, and P. Bożek, Time-dependent mean-field description for multiple electron transfer in slow ion-cluster collisions, Phys. Rev. A 57, R3165 (1998).
  75. K. Sekizawa and K. Yabana, Time-dependent Hartree-Fock calculations for multinucleon transfer processes in Ca40,48+Sn124, Ca40+Pb208, and 58Ni+Pb208 reactions, Phys. Rev. C 88, 014614 (2013).
  76. B. B. Back, H. Esbensen, C. L. Jiang, and K. E. Rehm, Recent developments in heavy-ion fusion reactions, Rev. Mod. Phys. 86, 317 (2014).
  77. J. Zhao, J. Xiang, Z.-P. Li, T. Nikšić, D. Vretenar, and S.-G. Zhou, Time-dependent generator-coordinate-method study of mass-asymmetric fission of actinides, Phys. Rev. C 99, 054613 (2019).
  78. K. Sekizawa and K. Hagino, Time-dependent Hartree-Fock plus Langevin approach for hot fusion reactions to synthesize the Z=120 superheavy element, Phys. Rev. C 99, 051602(R) (2019).
  79. G. Carleo, Y. Nomura, and M. Imada, Constructing exact representations of quantum many-body systems with deep neural networks, Nat. Commun. 9, 5322 (2018).
  80. K. Hashimoto, S. Sugishita, A. Tanaka, and A. Tomiya, Deep learning and the AdS/CFT correspondence, Phys. Rev. D 98, 046019 (2018).
  81. K. Hashimoto, S. Sugishita, A. Tanaka, and A. Tomiya, Deep learning and holographic QCD, Phys. Rev. D 98, 106014 (2018).
  82. K. Hashimoto, AdS/CFT correspondence as a deep Boltzmann machine, Phys. Rev. D 99, 106017 (2019).
  83. T. Ozaki, K. Hino, H. Kawai, and M. Toyoda, ADPACK Ver. 2.2, https://www.openmx-square.org/adpack_man2.2/adpack2_2.html.

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Phys. Rev. Research 5, 033189– Published 14 September, 2023
  • Received 19 February 2023
  • Accepted 9 July 2023
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